Borel-Cantelli Thresholds and Algorithmic Randomness in Halting Probability — E8 Intelligence Research

FINDING: Borel-Cantelli lemmas provide the threshold condition for almost-sure occurrence of infinitely many events, linking probability sums to binary/exponential decay rates. | MATH: First lemma: If Σₖ P(Aₖ) < ∞, then P(limsup Aₖ) = 0. Second lemma (independence): If Σₖ P(Aₖ) = ∞, then P(limsup Aₖ) = 1. Critical boundary: Σₖ 2⁻ᵏ converges (geometric series, ratio 1/2). The halting probability Ω = Σₖ 2⁻|pₖ| (Chaitin) is a normal number whose bits are algorithmically random — each bit deviation from 0.5 decays only as ~1/k, not exponentially, because Ω's bits are incompressible. | CONNECTION: The geometric series ratio 1/2 is the binary analogue of the golden ratio's self-similarity — both define convergent self-similar structures. The critical exponent k in 2⁻ᵏ maps to the dyadic lattice (Z[1/2]), a 1-dimensional analogue of crystallographic root systems. The threshold Σₖ P(Aₖ) = ∞ vs < ∞ is a phase transition at the "edge of convergence" — analogous to the critical point in percolati Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com